263.] ELLIPSOIDS OF INERTIA. 



262. To make use of this form for /, the principal axes at the point 

 O, i.e. the axes of the momental ellipsoid (10), must be known. The 

 determination of the axes of an ellipsoid whose equation referred to the 

 centre is given is a well-known problem of analytic geometry. It can 

 be solved by considering that the semi-axes are those radii vectores of 

 the surface that are normal to it. The direction cosines of the normal 

 of any surface F(x, y, z) = o are proportional to the partial derivatives 

 dF/dx, dF/dy, dF/dz. If, therefore, the radius vector p is a semi- 

 axis, its direction-cosines a, ft, y must be proportional to the partial 

 derivatives of (10) ; i.e. we must have 



Cz 



or dividing the numerators by p, 



Aa-F(3Ey = - Fa + Bft - Dy __ - Ea - Dft + Cy 

 a (3 y 



Denoting the common value of these fractions by /, we have 



al= Aa Fft - Ey, ftl= - Fa + Bft - Dy, y/= - Ea - Dft + Cy ; 



multiplying these equations by a, ft, y, and adding, we find 



/ = A a 2 + B^ + Cy 2 - 2 Dfty 2 Eya 2 Faft^ . 



which, compared with (9), shows that / is the moment of inertia for 

 the axis (a, ft, y) . To obtain it in function of A, B, C, D, E, F, we 

 write the preceding three equations in the form 



(S-A)a + Fft+ Ey=o, 



>y = o, (13) 



whence, eliminating a, ft, y, we find / determined by the cubic equation 

 I -A, F, E 



F, I-B, D 



= o. (14) 



E, D, I- C 



The roots of this cubic are the three principal moments /i, / 2 , / 3 of the 

 point O. The direction-cosines of the principal axes are then found by 

 substituting successively I l9 7 2 , 7 3 in (13) and solving for a, ft, y. 



263. The geometrical representation of the moments of 

 inertia for all lines passing through a point by means of the 



